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Theorem bj-ralcom4 30995
Description: Remove from ralcom4 3077 dependency on ax-ext 2380 and ax-13 2026 (and on df-or 368, df-an 369, df-tru 1408, df-sb 1764, df-clab 2388, df-cleq 2394, df-clel 2397, df-nfc 2552, df-v 3060). This proof uses only df-ral 2758 on top of first-order logic. (Contributed by BJ, 13-Jun-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-ralcom4  |-  ( A. x  e.  A  A. y ph  <->  A. y A. x  e.  A  ph )
Distinct variable groups:    x, y    y, A
Allowed substitution hints:    ph( x, y)    A( x)

Proof of Theorem bj-ralcom4
StepHypRef Expression
1 19.21v 1752 . . . . 5  |-  ( A. y ( x  e.  A  ->  ph )  <->  ( x  e.  A  ->  A. y ph ) )
21bicomi 202 . . . 4  |-  ( ( x  e.  A  ->  A. y ph )  <->  A. y
( x  e.  A  ->  ph ) )
32albii 1661 . . 3  |-  ( A. x ( x  e.  A  ->  A. y ph )  <->  A. x A. y
( x  e.  A  ->  ph ) )
4 alcom 1869 . . 3  |-  ( A. x A. y ( x  e.  A  ->  ph )  <->  A. y A. x ( x  e.  A  ->  ph ) )
53, 4bitri 249 . 2  |-  ( A. x ( x  e.  A  ->  A. y ph )  <->  A. y A. x
( x  e.  A  ->  ph ) )
6 df-ral 2758 . 2  |-  ( A. x  e.  A  A. y ph  <->  A. x ( x  e.  A  ->  A. y ph ) )
7 df-ral 2758 . . 3  |-  ( A. x  e.  A  ph  <->  A. x
( x  e.  A  ->  ph ) )
87albii 1661 . 2  |-  ( A. y A. x  e.  A  ph  <->  A. y A. x ( x  e.  A  ->  ph ) )
95, 6, 83bitr4i 277 1  |-  ( A. x  e.  A  A. y ph  <->  A. y A. x  e.  A  ph )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184   A.wal 1403    e. wcel 1842   A.wral 2753
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-11 1866
This theorem depends on definitions:  df-bi 185  df-ex 1634  df-ral 2758
This theorem is referenced by: (None)
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